3.3: Absolute Value

Overview

• Geometric Definition of Absolute Value
• Algebraic Definition of Absolute Value

Geometric Definition of Absolute Value

Absolute Value - Geometric Approach

The absolute value of a number $a$, denoted $|a|$, is the distance fromn $a$ to $0$ on the number line.

Absolute value speaks to the question of "how far," and not "which way." The phrase how far implies length, and length is always a nonnegative (zero or positive) quantity. Thus, the absolute value of a number is a nonnegative number. This is shown in the following examples:

Example $\PageIndex{1}$

$|4| = 4$

Example $\PageIndex{2}$

$|-4| = 4$

Example $\PageIndex{3}$

$|0| = 0$

Example $\PageIndex{4}$

$-|5| = -5$

The quantity on the left side of the equal sign is read as "negative the absolute value of $5$." The absolute value of $5$ is $5$. Hence, negative the absolute value of $5$ is $−5$.

Example $\PageIndex{5}$

$-|-3| = -3$

The quantity on the left side of the equal sign is read as "negative the absolute value of $−3$." The absolute value of $−3$ is $3$. Hence, negative the absolute value of $−3$ is $−(3)=−3$.

Algebraic Definition of Absolute Value

The problems in the first example may help to suggest the following algebraic definition of absolute value. The definition is interpreted below. Examples follow the interpretation.

Absolute Value-Algebraic Approach

The absolute value of a number $a$ is

$|a|=\left\{\begin{aligned} a & \text { if } a \geq 0 \\-a & \text { if } a<0 \end{aligned}\right.$

The algebraic definition takes into account the fact that the number $a$ could be either positive or zero ($\ge 0)$ or negative ($<0$).

1. If the number a is positive or zero ($\ge 0$), the first part of the definition applies. The first part of the definition tells us that if the number enclosed in the absolute bars is a nonnegative number, the absolute value of the number is the number itself.
2. If the number a is negative ($<0$), the second part of the definition applies. The second part of the definition tells us that if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. The opposite of a negative number is a positive number.

Sample Set A

Use the algebraic definition of absolute value to find the following values.

Example $\PageIndex{6}$

$|8|$

The number enclosed within the absolute value bars is a nonnegative number so the first part of the definition applies. This part says that the absolute value of $8$ is $8$ itself.

Example $\PageIndex{7}$

$|-3|$

The number enclosed within absolute value bars is a negative number so the second part of the definition applies. This part says that the absolute value of $−3$ is the opposite of $−3$, which is $−(−3)$. By the double-negative property, $−(−3)=3$.

Practice Set A

Use the algebraic definition of absolute value to find the following values.

Exercise $\PageIndex{1}$

$|7|$

Answer

$7$

Exercise $\PageIndex{2}$

$|9|$

Answer

$9$

Exercise $\PageIndex{3}$

$|-12|$

Answer

$12$

Exercise $\PageIndex{4}$

$|-5|$

Answer

$5$

Exercise $\PageIndex{5}$

$-|8|$

Answer

$-8$

Exercise $\PageIndex{6}$

$-|1|$

Answer

$-1$

Exercise $\PageIndex{7}$

$-|-52|$

Answer

$-52$

Exercise $\PageIndex{8}$

$-|-31|$

Answer

$-31$

Exercises

For the following problems, determine each of the values.

Exercise $\PageIndex{1}$

$|5|$

Answer

$5$

Exercise $\PageIndex{2}$

$|3|$

Exercise $\PageIndex{3}$

$|6|$

Answer

$6$

Exercise $\PageIndex{4}$

$|14|$

Exercise $\PageIndex{5}$

$|-8|$

Answer

$8$

Exercise $\PageIndex{6}$

$|-10|$

Exercise $\PageIndex{7}$

$|-16|$

Answer

$16$

Exercise $\PageIndex{8}$

$-|8|$

Exercise $\PageIndex{9}$

$-|12|$

Answer

$-12$

Exercise $\PageIndex{10}$

$-|47|$

Exercise $\PageIndex{11}$

$-|9|$

Answer

$-9$

Exercise $\PageIndex{12}$

$|-9|$

Exercise $\PageIndex{13}$

$|-1|$

Answer

$1$

Exercise $\PageIndex{14}$

$|-4|$

Exercise $\PageIndex{15}$

$-|3|$

Answer

$-3$

Exercise $\PageIndex{16}$

$-|7|$

Exercise $\PageIndex{17}$

$-|-14|$

Answer

$-14$

Exercise $\PageIndex{18}$

$-|-19|$

Exercise $\PageIndex{19}$

$-|-28|$

Answer

$-28$

Exercise $\PageIndex{20}$

$-|-31|$

Exercise $\PageIndex{21}$

$-|-68|$

Answer

$-68$

Exercise $\PageIndex{22}$

$|0|$

Exercise $\PageIndex{23}$

$|-26|$

Answer

$26$

Exercise $\PageIndex{24}$

$-|-26|$

Exercise $\PageIndex{25}$

$-|-(-8)|$

Answer

$-8$

Exercise $\PageIndex{26}$

$-|-(-4)|$

Exercise $\PageIndex{27}$

$-|-(-1)|$

Answer

$-1$

Exercise $\PageIndex{28}$

$-|-(-7)|$

Exercise $\PageIndex{29}$

$-(-|4|)$

Answer

$4$

Exercise $\PageIndex{30}$

$-(-|2|)$

Exercise $\PageIndex{31}$

$-(-|-6|)$

Answer

$6$

Exercise $\PageIndex{32}$

$-(-|-42|)$

Exercise $\PageIndex{33}$

$|-|-3||$

Answer

$3$

Exercise $\PageIndex{34}$

$|-|-15||$

Exercise $\PageIndex{35}$

$|-|-12||$

Answer

$12$

Exercise $\PageIndex{36}$

$|-|-29||$

Exercise $\PageIndex{37}$

$6 - |-2||$

Answer

$4$

Exercise $\PageIndex{38}$

$|18 - |-11||$

Exercise $\PageIndex{39}$

$|5 - |-1||$

Answer

$4$

Exercise $\PageIndex{40}$

$|10 - |-3||$

Exercise $\PageIndex{41}$

$|-(17 - |-12|)|$

Answer

$5$

Exercise $\PageIndex{42}$

$|-(46 - |-24|)|$

Exercise $\PageIndex{43}$

$|5| - |-2|$

Answer

$3$

Exercise $\PageIndex{44}$

$|-2|^3$

Exercise $\PageIndex{45}$

$|-(2 \cdot 3)|$

Answer

$6$

Exercise $\PageIndex{46}$

$|-2| + |-9|$

Exercise $\PageIndex{47}$

$|-6| + |4|)^2$

Answer

$100$

Exercise $\PageIndex{48}$

$|-1| - |1|)^3$

Exercise $\PageIndex{49}$

$(|4| + |-6|)^2) - (|-2|)^3$

Answer

$92$

Exercise $\PageIndex{50}$

$-[|-10| -6]^2$

Exercise $\PageIndex{51}$

$-[-(-|-4|+|-3|)^3]^2$

Answer

$-1$

Exercise $\PageIndex{52}$

A Mission Control Officer at Cape Canaveral makes the statement "lift-off, $T$ minus $50$ seconds." How long before lift-off?

Exercise $\PageIndex{53}$

Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $2,400,000. Use absolute value notation to describe this company's debt.

Answer

$|$ -2,400,600|$

Exercise $\PageIndex{54}$

A particular machine is set correctly if upon action its meter reads $0$ units. One particular machine has a meter reading of $−1.6$ upon action. How far is this machine off its correct setting?

Exercises for Review

Exercise $\PageIndex{55}$

Write the following phrase using algebraic notation: "four times $(a+b)$."

Answer

$4(a + b)$

Exercise $\PageIndex{56}$

Is there a smallest natural number? If so, what is it?

Exercise $\PageIndex{57}$

Name the property of real numbers that makes $5+a=a+5$ a true statement.

Answer

commutative property of addition

Exercise $\PageIndex{58}$

Find the quotient of $\dfrac{x^6y^8}{x^4y^3}$

Exercise $\PageIndex{59}$

Simplify $-(-4)$

Answer

$4$